The 4,300-seat performance space about an hour north of Carnegie Hall was eerily empty, except for nine judges in uniform sitting behind a thick black curtain.

Ada Brooks, her mouth dry from nerves, lifted the bell of her euphonium, a smaller relative of the tuba, and prepared to play the notes that could determine her future.

“Breathe,” she thought. “The beginnings are the most treacherous part.”

Ms. Brooks had told herself this before. Her fervent pursuit to professionally play the euphonium, which is not used in traditional symphony orchestras, had come with many stressful auditions. This one was her 10th for the institution that calls itself the nation’s largest employer of musicians: the United States military.

Time and time again she had practiced and prepared and tried to remember to breathe. She was turned down repeatedly or offered jobs in regional bands. Now came an opportunity for a premium position, a rarely open seat in the prestigious West Point Band.

Some aspects of the audition — like playing for a jury hidden behind a curtain, to guard against potential bias — would be familiar to most orchestra musicians. Others were unique to the military. Two of the other four candidates said they had to lose weight to qualify, and the finalists were tested for coordination in marching drills.

Scores of regional military bands represent the armed forces at ceremonies, parades and holiday celebrations. About a dozen premier bands, including the U.S. Military Academy’s ensemble in West Point, N.Y., perform at inaugurations and foreign dignitary visits.